Time evolution of the complexity in chaotic systems: concrete examples

TL;DR

This work challenges the view that bi-invariant complexity cannot display the canonical chaotic-growth behavior by analyzing the SYK model under a fixed total energy. Using Nielsen's complexity geometry with a non-Riemannian (Finsler), bi-invariant metric, the authors demonstrate linear growth up to $t \sim e^{N}$, followed by saturation and small fluctuations, and show that Lloyd's bound is realized. They argue that unitary invariance enforces left/right complexity equivalence, making bi-invariant complexity a natural candidate for QM/QFT complexity, and that the $p=1$ case uniquely preserves a finite growth rate at large $N$. The results emphasize that fixing total energy, rather than coupling, is crucial for obtaining the expected exponential-time growth, and they highlight the fundamental role of non-Riemannian Finsler geometry in this context.

Abstract

We investigate the time evolution of the complexity of the operator by the Sachdev-Ye-Kitaev (SYK) model with $N$ Majorana fermions. We follow Nielsen's idea of complexity geometry and geodesics thereof. We show that it is possible that the bi-invariant complexity geometry can exhibit the conjectured time evolution of the complexity in chaotic systems: i) linear growth until $t\sim e^{N}$, ii) saturation and small fluctuations after then. We also show that the Lloyd's bound is realized in this model. Interestingly, these characteristic features appear only if the complexity geometry is the most natural "non-Riemannian" Finsler geometry. This serves as a concrete example showing that the bi-invariant complexity may be a competitive candidate for the complexity in quantum mechanics/field theory (QM/QFT). We provide another argument showing a naturalness of bi-invariant complexity in QM/QFT. That is that the bi-invariance naturally implies the equivalence of the right-invariant complexity and left-invariant complexity, either of which may correspond to the complexity of a given operator. Without bi-invariance, one needs to answer why only right (left) invariant complexity corresponds to the "complexity", instead of only left (right) invariant complexity.

Figures (5)

• Figure 1: A continuous (purple) curve $c(s)$ connects the identity ($c(0)=\hat{\mathbb{I}}$) and a particular operator $\hat{O}$ ($c(1)=\hat{O}$). It can be approximated by a discrete form (blue lines), where every intermediate point ($\hat{O}_n$) is also an operator.
• Figure 2: The relationship between $\Delta E_{\min}$ and $N$. The red line is the fitting curve $\Delta E_{\min}/\mathcal{J}=0.18N^{-1}+0.87N^{-2}$.
• Figure 3: Left panel: the complexity growth when $N=16,18,20,22$ and $N\rightarrow\infty$ with $\langle E \rangle =1$. Right panel: the critical time $t_c$ vs the fermion number $N$ with a fixed $\mathcal{J}=1$. The red line is the fitting curve, which shows that $1/t_c|_{\mathcal{J}=1}\approx 0.01N+0.0265$.
• Figure 4: Left panel: relationship between the maximal complexity $\mathcal{C}_{\max}$ and the fermion number $N$. Right panel: relationship between the critical complexity $\mathcal{C}_{c}$ and the fermion number $N$.
• Figure 5: As two different costs describe the same underlying physical dynamics with a unique concept of complexity, there must be a "translator", which only depend on the complexity theory, not on individual systems. However, such a translator exists only if the complexity geometries ($\tilde{F}_{r(l)}$) are bi-invariant.